Refractometry:

Theory

The speed of light in a vacuum is always the same, but
when light moves through any other medium it travels more slowly since
it is constantly being absorbed and reemitted by the atoms in the material.
The ratio of the speed of light in a vacuum to the speed of light in another
substance is defined as the index of refraction (aka
refractive index or n) for
the substance.

Figure 1. Light crossing
from any transparent medium into another in which it has a different
speed, is refracted, i.e., bent from its original path (except when
the direction of travel is perpendicular to the boundary). In the
case shown, the speed of light in medium A is greater than the speed
of light in medium B.

Whenever light changes speed as it crosses a boundary
from one medium into another its direction of travel also changes, i.e.,
it is refracted (Figure 1). (In the special case of the light traveling
perpendicular to the boundary there is no change in direction upon entering
the new medium.) The relationship between light's speed in the two mediums
(vA and vB), the angles
of incidence (qA)
and refraction (qB)
and the refractive indexes of the two mediums (nA
and nB) is shown below:

Thus, it is not necessary to measure the speed of light
in a sample in order to determine its index of refraction. Instead, by
measuring the angle of refraction, and knowing the index of refraction
of the layer that is in contact with the sample, it is possible to determine
the refractive index of the sample quite accurately. Nearly all refractometers
utilize this principle, but may differ in their optical design.

Figure 2. Cross section
of part of the optical path of an Abbe refractometer. The sample
thickness has been exaggerated for clarity.

In the Abbe' refractometer the liquid sample is sandwiched
into a thin layer between an illuminating prism and a refracting prism
(Figure 2). The refracting prism is made of a glass with a high refractive
index (e.g., 1.75) and the refractometer is designed to be used with samples
having a refractive index smaller than that of the refracting prism. A
light source is projected through the illuminating prism, the bottom surface
of which is ground (i.e., roughened like a ground-glass joint), so each
point on this surface can be thought of as generating light rays traveling
in all directions. Inspection of Figure 2 shows that light traveling from
point A to point B will have the largest angle of incidence (qi)
and hence the largest possible angle of refraction (qr)
for that sample. All other rays of light entering the refracting prism
will have smaller qr
and hence lie to the left of point C. Thus, a detector placed on the back
side of the refracting prism would show a light region to the left and
a dark region to the right.

Samples with different refractive indexes will produce
different angles of refraction (see Equation 2 above and recall that the
angle of incidence and the refractive index of the prism are fixed) and
this will be reflected in a change in the position of the borderline between
the light and dark regions. By appropriately calibrating the scale, the
position of the borderline can be used to determine the refractive index
of any sample. In an actual Abbe' refractometer there is not a detector
on the back of the refracting prism, and there are additional optics,
but this is the essential principle.

In most liquids and solids the speed of light, and hence
the index of refraction, varies significantly with wavelength. (This variation
is referred to as dispersion, and it is what causes white
light moving through a prism to be refracted into a rainbow. Shorter wavelengths
are normally refracted more than longer ones.) Thus, for the most accurate
measurements it is necessary to use monochromatic light. The most widely
used wavelength of light for refractometry is the sodium D line at 589
nm.

If white light were used in the simple Abbe' refractometer
optics shown in Figure 2, dispersion would result in the light and dark
borderline being in different places for different wavelengths of light.
The resulting "fuzziness" of the borderline would make precise
work impossible. However, many Abbe' refractometers are able to operate
satisfactorily with white light by introducing a set of "compensating
prisms" into the optical path after the refracting prism. These compensating
prisms are designed so that they can be adjusted to correct (i.e., compensate
for) the dispersion of the sample in such a way that they reproduce the
refractive index that would be obtained with monochromatic light of 589
nm, the sodium D line.

As mentioned earlier, the speed of light in a substance
is slower than in a vacuum since the light is being absorbed and reemitted
by the atoms in the sample. Since the density of a liquid usually decreases
with temperature, it is not surprising that the speed of light in a liquid
will normally increase as the temperature increases. Thus, the
index of refraction normally decreases as the temperature increases
for a liquid (Table 1). For many organic liquids the index of refraction
decreases by approximately 0.0005 for every 1 °C increase in temperature.
However for water the variation is only about -0.0001/°C.

Many refractometers are equipped with a thermometer
and a means of circulating water through the refractometer to maintain
a given temperature. Most of the refractive index measurements reported
in the literature are determined at 20 or 25 °C.